Statistics of random processes. 1: General theory. Transl. from the Russian by A. B. Aries. Transl. Ed. : Stephen S. Wilson. 2nd rev. and exp. ed.

*(English)*Zbl 1008.62072
Applications of Mathematics. 5. Berlin: Springer. xv, 427 p. (2001).

For several reasons stochastic analysis has been among the scientific hits of the last few decades. The present authors not only made great contributions, but they wrote their “Statistics of Random Processes” which became one of the most influential books in this area. Its first edition in Russian appeared in 1974, see the review Zbl 0279.60021, and was soon translated into English in two volumes; for Vol. I from 1977 see the review Zbl 0364.60004 and for Vol. II from 1978 see Zbl 0369.60001.

Now under review is this second edition, once again in two volumes. The authors keep the whole content of the first edition but as expected, they have made several additions reflecting recent developments. Each chapter is updated, two new sub-sections are added, one on the Kalman filter under wrong initial conditions and another one on asymptotically optimal filtering under diffusion approximations. Useful comments are added to each chapter on recent progress including some contemporary applications such as stochastic finance. The reference list is also updated with essential recently published papers and books.

Thus the comprehensive content and the masterly written text make the book attractive for researchers in stochastic analysis and its applications as well as for graduate students in the area. The prediction of this reviewer is that this book will continue to be among the most useful and popular books on the subject in the decades to come.

Now under review is this second edition, once again in two volumes. The authors keep the whole content of the first edition but as expected, they have made several additions reflecting recent developments. Each chapter is updated, two new sub-sections are added, one on the Kalman filter under wrong initial conditions and another one on asymptotically optimal filtering under diffusion approximations. Useful comments are added to each chapter on recent progress including some contemporary applications such as stochastic finance. The reference list is also updated with essential recently published papers and books.

Thus the comprehensive content and the masterly written text make the book attractive for researchers in stochastic analysis and its applications as well as for graduate students in the area. The prediction of this reviewer is that this book will continue to be among the most useful and popular books on the subject in the decades to come.

Reviewer: Jordan M.Stoyanov (Newcastle upon Tyne)